Combined Variance of Two Groups

• For Set $$S$$: Mean $$\bar{x}_1 = \frac{n(a) + n(-a)}{2n} = 0$$
• For Set $$S'$$: Mean $$\bar{x}_2 = \frac{n(b) + n(-b)}{2n} = 0$$
• For Combined Set ($$S \cup S'$$): Combined Mean $$\bar{x} = \frac{2n(0) + 2n(0)}{4n} = 0$$
The formula for variance when the mean is zero is $$\sigma^2 = \frac{\sum x_i^2}{N}$$.
• For Set $$S$$:
$$\sigma_1^2 = \frac{n(a)^2 + n(-a)^2}{2n} = \frac{2na^2}{2n} = a^2 \implies a^2 = \sigma_1^2$$
• For Set $$S'$$:
$$\sigma_2^2 = \frac{n(b)^2 + n(-b)^2}{2n} = \frac{2nb^2}{2n} = b^2 \implies b^2 = \sigma_2^2$$

Using the same zero-mean variance formula for all $$4n$$ items:
$$\sigma_{combined}^2 = \frac{\sum_{S} x_i^2 + \sum_{S'} x_i^2}{4n}$$
$$\sigma_{combined}^2 = \frac{2na^2 + 2nb^2}{4n} = \frac{2n(a^2 + b^2)}{4n} = \frac{a^2 + b^2}{2}$$
Substitute $$\sigma_1$$ and $$\sigma_2$$ back into the formula.
$$\sigma_{combined}^2 = \frac{\sigma_1^2 + \sigma_2^2}{2}$$
$$\sigma_{combined} = \sqrt{\frac{\sigma_1^2 + \sigma_2^2}{2}}$$